The genesis of these coherent structures was found to be in the stability characteristics of the flow: in jets and shear-layers, the presence of an inflection point in the base flow leads to the appearance of an inviscid instability, namely the Kelvin-Helmholtz (KH) mode [1]}, [2]}. This mode extracts energy from the mean flow, resulting in the highly amplified structures observed experimentally [3]}, [4]}. This clear instability mechanism inspired the development of several modelling methods to predict the key features of wavepackets. While the application of locally parallel linear stability analysis resulted in only qualitative agreement with experimentally educed KH modes [5]}, [6]}, models that consider the jet development were shown to quantitatively capture the overall behaviour of these waves. For instance, [7]} proposed a theoretical method based on a WKB approximation to overcome the limitations of the local analysis, obtaining the solution for a slowly-varying mean flow. The parabolised stability equations (PSE), introduced by [8]}, improved upon the WKB approach by tracking the slow downstream evolution of the shape function and wavenumber [9]}, [10]}. Application of PSE to predict wavepacket shapes and phase velocities was first performed by [11]} for subsonic jets, which led to good agreement with modes extracted from data via spectral proper orthogonal decomposition (SPOD, [12]}) for frequencies relevant to jet noise. This comparison has since been extended to higher frequencies and azimuthal wavenumbers [13]}, ideally expanded supersonic jets [14]}, and jets with azimuthal non-homogeneity [15]}.
